The liberated electrons that do not leave a grain impart their energy through collisions to the lattice (Figure 7.5). Most of the ultraviolet radiation, therefore, serves to raise the grain temperature Td. Somewhat more dust heating is provided by the stronger flux of optical photons, which also excite internal electrons. Counting only this visible component, the total dust heating rate per unit volume is
Here the grain absorption efficiency Qv,abs is proportional to v in the visual regime. As before, we model Jv as arising from a diluted blackbody at the characteristic temperature T and peak frequency vmax. Equation (7.19) then becomes where W is the dilution factor. We define the nondimensional variable x = hv/kB T and find
The nondimensional integral has the numerical value 24.9. We next use the relation between ndad and nH, and set QVmax equal to 0.1, the value appropriate for the optical peak frequency
of vmax = 3 x 1014 s 1. (Recall equation (2.40) and Figure 2.10.) The expression for then becomes
We emphasize that this relation governs the heating of the dust grains, not the gas. The grains can transfer energy to the gas only at cloud densities high enough for good collisional coupling between the two components.
We have not yet mentioned the effect of interstellar radiation on the dominant constituent of star-forming clouds, molecular hydrogen itself. As we saw in Chapter 5, the absorption of a photon with energy hv greater than 11.2 eV promotes H2 to an excited electronic state. Most often, the excited molecule drops to the electronic ground state and then cascades down the rovibrational levels within that state. In quiescent molecular clouds, this decay occurs through the emission of ultraviolet and infrared photons, but the energy can be given to other colliding species at the high densities and temperatures behind shocks. If the excited H2 instead dissociates, it emits a photon of energy hv — AEdiss - e and imparts kinetic energy e to the separate atoms. On average, e is about 2 eV. This energy quickly spreads into the gas through collisions.
In a quiescent cloud of molecular hydrogen exposed to the full interstellar radiation field, this secondary effect of H2 dissociation would completely dominate the heating. As we have already noted, however, such a cloud cannot exist. Molecular hydrogen is only found at a depth below the cloud surface where the ultraviolet flux has already been severely attenuated by excitation and dissociation of the outer H2 molecules and by dust absorption. Consequently, heating by dissociation is relegated to a minor role.
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